Strongly Stable Generalized Finite Element Method (SSGFEM) for a non-smooth interface problem
In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner. The SSGFEM employs enrichments of 2 types: the nodes in a neighborhood of the corner are enriched by singular functions characterizing the sing...
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Published in | Computer methods in applied mechanics and engineering Vol. 344; pp. 538 - 568 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Amsterdam
Elsevier B.V
01.02.2019
Elsevier BV |
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Abstract | In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner. The SSGFEM employs enrichments of 2 types: the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution, while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface. Thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions. Both types of enrichments have been modified by a simple local procedure of “subtracting the interpolant.” A simple local orthogonalization technique (LOT) also has been used at the nodes enriched with both enrichment functions. We prove that the SSGFEM yields the optimal order of convergence. The numerical experiments presented in this paper indicate that the conditioning of the SSGFEM is not worse than that of the standard finite element method, and the conditioning is robust with respect to the position of the mesh relative to the interface.
•GFEM for 2D non-smooth interface problem.•Singular enrichment functions in addition to distance based enrichment functions.•Proof of optimal convergence of the GFEM.•Experimental study of conditioning and robustness of GFEM.•The notion of Strongly Stable GFEM (SSGFEM). |
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AbstractList | In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner. The SSGFEM employs enrichments of 2 types: the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution, while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface. Thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions. Both types of enrichments have been modified by a simple local procedure of “subtracting the interpolant.” A simple local orthogonalization technique (LOT) also has been used at the nodes enriched with both enrichment functions. We prove that the SSGFEM yields the optimal order of convergence. The numerical experiments presented in this paper indicate that the conditioning of the SSGFEM is not worse than that of the standard finite element method, and the conditioning is robust with respect to the position of the mesh relative to the interface.
•GFEM for 2D non-smooth interface problem.•Singular enrichment functions in addition to distance based enrichment functions.•Proof of optimal convergence of the GFEM.•Experimental study of conditioning and robustness of GFEM.•The notion of Strongly Stable GFEM (SSGFEM). In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner. The SSGFEM employs enrichments of 2 types: the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution, while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface. Thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions. Both types of enrichments have been modified by a simple local procedure of “subtracting the interpolant.” A simple local orthogonalization technique (LOT) also has been used at the nodes enriched with both enrichment functions. We prove that the SSGFEM yields the optimal order of convergence. The numerical experiments presented in this paper indicate that the conditioning of the SSGFEM is not worse than that of the standard finite element method, and the conditioning is robust with respect to the position of the mesh relative to the interface. |
Author | Banerjee, Uday Zhang, Qinghui Babuška, Ivo |
Author_xml | – sequence: 1 givenname: Qinghui surname: Zhang fullname: Zhang, Qinghui email: zhangqh6@mail.sysu.edu.cn organization: Guangdong Province Key Laboratory of Computational Science and School of Data and Computer Science, Sun Yat-Sen University, Guangzhou, 510006, PR China – sequence: 2 givenname: Uday surname: Banerjee fullname: Banerjee, Uday email: banerjee@syr.edu organization: Department of Mathematics, 215 Carnegie, Syracuse University, Syracuse, NY 13244, USA – sequence: 3 givenname: Ivo surname: Babuška fullname: Babuška, Ivo organization: ICES, University of Texas at Austin, Austin, TX 78712, USA |
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Snippet | In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner.... |
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SubjectTerms | Conditioning Convergence Enrichment Finite element analysis Finite element method GFEM/XFEM Interface Mathematical analysis Nodes Nonlinear programming Robustness (mathematics) Scaled condition number Singularity SSGFEM |
Title | Strongly Stable Generalized Finite Element Method (SSGFEM) for a non-smooth interface problem |
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